Limit theorems for logarithmic means
The author examines his logarithmic means \[ L_ r(x_ 0,...,x_ n) = \begin{cases} (\int_{A_ n}n!(\sum^{n}_{k=0}x_ kv_ k)^ rdv_ 0...dv_{n-1})^{1/r} & (r\neq 0), \\ \exp\int_{A_ n}n! \log(\sum^{n}_{k=0} x_ kv_ k)dv_ 0... dv_{n-1} & (r=0), \end{cases} \] where \(A_ n=\{(v_ 0,...,v_{n-1}):\) \(0\leq v_ k\leq 1\); \(k=0,...,n-1\); \(0\leq \sum^{n-1}_{k=0}v_ k\leq 1\}\) and \(v_ n:=1- \sum^{n-1}_{k=0}v_ k\), their limits and possible applications to probability theory. Similar computations are done for the Leach-Sholander means [\textit{E. B. Leach} and \textit{C. Sholander}, J. Math. Anal. Appl. 104, 390-407 (1984; Zbl 0558.26014)]: \[ \begin{multlined} E(r,s;x_ 0,...,x_ n)= \\ (\int_{A_ n}(\sum^{n-1}_{k=0}x_ kv_ k)^{s-n}dv_ 0...dv_{n-1}/\int_{A_ n}(\sum^{n-1}_{k=0}x_ kv_ k)^{r- n}dv_ 0...dv_{n-1})^{1/(s-r)}\qquad (r\neq s),\end{multlined} \] \[ \begin{multlined} E(r,r;x_ 0,...,x_ n)= \\ \exp (\int_{A_ n}(\sum^{n}_{k=0}x_ kv_ k)^{r-n}\log (\sum^{n}_{k=0}x_ kv_ k)dv_ 0...dv_{n-1}/\int_{A_ n}(\sum^{n}_{k=0}x_ kv_ k)^{r-n}dv_ 0...dv_{n-1}),\qquad (r=s). \end{multlined} \]
scientific article; zbMATH DE number 4002423
scientific article